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Theorem llynlly 23501
Description: A locally 𝐴 space is n-locally 𝐴: the "n-locally" predicate is the weaker notion. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llynlly (𝐽 ∈ Locally 𝐴𝐽 ∈ 𝑛-Locally 𝐴)

Proof of Theorem llynlly
Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llytop 23496 . 2 (𝐽 ∈ Locally 𝐴𝐽 ∈ Top)
2 llyi 23498 . . . . 5 ((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) → ∃𝑢𝐽 (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
3 simpl1 1190 . . . . . . . 8 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Locally 𝐴)
43, 1syl 17 . . . . . . 7 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
5 simprl 771 . . . . . . 7 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢𝐽)
6 simprr2 1221 . . . . . . 7 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑦𝑢)
7 opnneip 23143 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑢𝐽𝑦𝑢) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
84, 5, 6, 7syl3anc 1370 . . . . . 6 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
9 simprr1 1220 . . . . . . 7 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢𝑥)
10 velpw 4610 . . . . . . 7 (𝑢 ∈ 𝒫 𝑥𝑢𝑥)
119, 10sylibr 234 . . . . . 6 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢 ∈ 𝒫 𝑥)
128, 11elind 4210 . . . . 5 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥))
13 simprr3 1222 . . . . 5 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝐽t 𝑢) ∈ 𝐴)
142, 12, 13reximssdv 3171 . . . 4 ((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴)
15143expb 1119 . . 3 ((𝐽 ∈ Locally 𝐴 ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴)
1615ralrimivva 3200 . 2 (𝐽 ∈ Locally 𝐴 → ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴)
17 isnlly 23493 . 2 (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
181, 16, 17sylanbrc 583 1 (𝐽 ∈ Locally 𝐴𝐽 ∈ 𝑛-Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086  wcel 2106  wral 3059  wrex 3068  cin 3962  wss 3963  𝒫 cpw 4605  {csn 4631  cfv 6563  (class class class)co 7431  t crest 17467  Topctop 22915  neicnei 23121  Locally clly 23488  𝑛-Locally cnlly 23489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-top 22916  df-nei 23122  df-lly 23490  df-nlly 23491
This theorem is referenced by:  llyssnlly  23502  efmndtmd  24125
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